Abstract
In this paper, we derive generalized Bern-Carrasco-Johansson (BCJ) relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM, which are derived from Feynman rules, Abelian Z-theory and Cachazo-He-Yuan (CHY) formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.
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Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
Y.-X. Chen, Y.-J. Du and B. Feng, A proof of the explicit minimal-basis expansion of tree amplitudes in gauge field theory, JHEP 02 (2011) 112 [arXiv:1101.0009] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE].
G. Chen and Y.-J. Du, Amplitude relations in non-linear σ-model, JHEP 01 (2014) 061 [arXiv:1311.1133] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
Q. Ma, Y.-J. Du and Y.-X. Chen, On primary relations at tree-level in string theory and field theory, JHEP 02 (2012) 061 [arXiv:1109.0685] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
Y.-J. Du and C.-H. Fu, Explicit BCJ numerators of nonlinear sigma model, JHEP 09 (2016) 174 [arXiv:1606.05846] [INSPIRE].
H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills amplitude relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, New identities among gauge theory amplitudes, Phys. Lett. B 691 (2010) 268 [arXiv:1006.3214] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Proof of gravity and Yang-Mills amplitude relations, JHEP 09 (2010) 067 [arXiv:1007.3111] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α ′ -corrections from the open string, JHEP 06 (2017) 093 [arXiv:1608.02569] [INSPIRE].
Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033 [arXiv:1703.05717] [INSPIRE].
S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].
D. Nandan, J. Plefka, O. Schlotterer and C. Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Relations for Einstein-Yang-Mills amplitudes from the CHY representation, Phys. Lett. B 767 (2017) 86 [arXiv:1607.06036] [INSPIRE].
O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills, JHEP 11 (2016) 074 [arXiv:1608.00130] [INSPIRE].
C.-H. Fu, Y.-J. Du, R. Huang and B. Feng, Expansion of Einstein-Yang-Mills amplitude, JHEP 09 (2017) 021 [arXiv:1702.08158] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Explicit formulae for Yang-Mills-Einstein amplitudes from the double copy, JHEP 07 (2017) 002 [arXiv:1703.00421] [INSPIRE].
F. Teng and B. Feng, Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame, JHEP 05 (2017) 075 [arXiv:1703.01269] [INSPIRE].
Y.-J. Du, B. Feng and F. Teng, Expansion of all multitrace tree level EYM amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
M. Kiermaier, Gravity as the square of gauge theory, talk at Amplitudes 2010, http://www.strings.ph.qmul.ac.uk/∼theory/Amplitudes2010/Talks/MK2010.pdf, Queen Mary University, London, U.K., May 2010.
Z. Bern, T. Dennen, Y.-T. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ relation of color scalar theory and KLT relation of gauge theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, An algebraic approach to BCJ numerators, JHEP 03 (2013) 050 [arXiv:1212.6168] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on construction of dual-trace factor in Yang-Mills theory, JHEP 10 (2013) 069 [arXiv:1305.2996] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, The construction of dual-trace factor in Yang-Mills theory, JHEP 07 (2013) 057 [arXiv:1304.2978] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on symmetric BCJ numerator, JHEP 08 (2014) 098 [arXiv:1403.6262] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills scattering amplitudes from scattering equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
L.A. Barreiro and R. Medina, RNS derivation of N -point disk amplitudes from the revisited S-matrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].
R.H. Boels and R. Medina, Graviton and gluon scattering from first principles, Phys. Rev. Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].
R.H. Boels and H. Lüo, A minimal approach to the scattering of physical massless bosons, JHEP 05 (2018) 063 [arXiv:1710.10208] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, On-shell recursion relations for effective field theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].
C. Cheung, C.-H. Shen and C. Wen, Unifying relations for scattering amplitudes, JHEP 02 (2018) 095 [arXiv:1705.03025] [INSPIRE].
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Du, YJ., Zhang, Y. Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes. J. High Energ. Phys. 2018, 177 (2018). https://doi.org/10.1007/JHEP07(2018)177
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DOI: https://doi.org/10.1007/JHEP07(2018)177